A Kruskal–Katona Type Theorem for the Linear Lattice

We present an analog of the well-known Kruskal–Katona theorem for the poset of subspaces of PG(n,2) ordered by inclusion. For givenk,ℓ (k < ℓ) andmthe problem is to find a family of sizemin the set of ℓ-subspaces of PG(n,2), containing the minimal number ofk-subspaces. We introduce two lexicographic type orders O1and O2on the set of ℓ-subspaces, and prove that the firstmof them, taken in the order O1, provide a solution in the casek = 0 and arbitrary ℓ > 0, and one taken in the order O2, provide a solution in the case ℓ = n − 1 and arbitraryk < n − 1. Concerning other values ofkand ℓ, we show that forn ≥ 3 the considered poset is not Macaulay by constructing a counterexample in the case ℓ = 2 andk = 1.